Signature pairs of positive polynomials

A well-known theorem of Quillen says that if $r(z,\bar{z})$ is a bihomogeneous polynomial on ${\mathbb{C}}^n$ positive on the sphere, then there exists $d$ such that $r(z,\bar{z}){\lVert z \rVert}^{2d}$ is a squared norm. We obtain effective bounds relating this $d$ to the signature of $r$. We obtain the sharp bound for $d=1$, and for $d > 1$ we obtain a bound that is of the correct order as a function of $d$ for fixed $n$. The current work adds to an extensive literature on positivity classes for real polynomials. The classes $\Psi_d$ of polynomials for which $r(z,\bar{z}){\lVert z \rVert}^{2d}$ is a squared norm interpolate between polynomials positive on the sphere and those that are Hermitian sums of squares.